Calculus

Problem 27301

Population de 33000 habitants, taux de naissance $3\%$ , mortalité $1\%$ , 220 départs/an. Trouvez $P(t)$ et $P(10)$ , puis le temps pour doubler.

See Solution

Problem 27302

Berechnen Sie die Fläche zwischen dem Graphen von $f(x)=-x^{2}-5x-4$ und der $x$ -Achse im Intervall $[-5, 0]$ .

See Solution

Problem 27303

A parallel-plate capacitor with 20 cm sides has an electric field increasing at $\frac{d E}{d t}=5.7 \times 10^{11} \mathrm{~V/m \cdot s}$ . Find the instantaneous current $(i)$ .

See Solution

Problem 27304

Calculate the integral $\int_{-1}^{1} (x^{2}-1) \, dx$ .

See Solution

Problem 27305

Berechne die Gesamtfläche zwischen dem Graphen von $f(x) = 0,5x^2 - 3x$ und der x-Achse.

See Solution

Problem 27306

Calculate the integral from -1 to 2 of the function $x^{2} - 1$ .

See Solution

Problem 27307

Find the integral of the constant function $f(x)=5$ . What is the result?

See Solution

Problem 27308

Find the integral $\int(3 x^{2}+4 x+2) \, dx$ . What is the result?

See Solution

Problem 27309

Evaluate the integral $\int\left(e^{x} \cos (x)\right) d x$ .

See Solution

Problem 27310

What is the correct notation for the indefinite integral of $f(x)$ with respect to $x$ ? Options: $\partial \partial f(x) d x$ , $\iint f(x) d x$ , $\partial f(x) d x$ , $\int f(x) d x$ .

See Solution

Problem 27311

Identify the definite integral from these options: $\int f(x) d x$ , $\iint f(x) d x$ , $\partial f(x) d x$ , $\int_{a}^{b} f(x) d x$ .

See Solution

Problem 27312

Find the integral of $x \sin (x)$ . Choose from: $\cos (x)+x \sin (x)+c$ , $\cos (x)-x \sin (x)+c$ , $\sin (x)-x \cos (x)+c$ , $\sin (x)+x \cos (x)+c$ .

See Solution

Problem 27313

Find the total distance traveled by a car with velocity $v(t)=3 t^{2}-6 t+4$ m/s from $t=1$ to $t=5$ : $\int_{1}^{5} v(t) dt$ .

See Solution

Problem 27314

A savings account has $\$ 500.00$ at $14\%$ continuous interest. How much can be spent on a bike after 3 years?

See Solution

Problem 27315

Find the rate of change of the curve $f(x)=-x^{3}$ from $x=0$ to $x=2$ . Options: -4, -2, 0, 2, 5.

See Solution

Problem 27316

Betty deposited \$1,000 in a savings account with 13% continuous interest. How much can she spend on a bike in 1 year?

See Solution

Problem 27317

Find when the velocity of the particle at $x(t)=(t+1)(t-3)^{3}$ is increasing. Choices: (A) $t>3$ (B) $t<1$ (C) $1<t<3$ (D) $t<1$ or $t>3$

See Solution

Problem 27318

Find the integral of $\sin^{2} x$ with respect to $x$ : $\int \sin^{2} x \, dx$ .

See Solution

Problem 27319

Estimate the instantaneous rate of change of height $h(t)=-16 \cos \left(\frac{2 \pi t}{32}\right)+18$ at $t=19$ s with $\Delta t=0.001$ . Round to one decimal.

See Solution

Problem 27320

Estimate the instantaneous rate of change of $h(t)=-16 \cos \left(\frac{2 \pi t}{32}\right)+18$ at $t=19$ with $\Delta t=0.001$ . Which statement is FALSE about average rate of change?

See Solution

Problem 27321

Find the slope of the tangent to the function $f(x)=-3 x^{2}+11 x-6$ at the point $P(-4,98)$ .

See Solution

Problem 27322

Find the instantaneous rate of change of height, $h(t)=-16 \cos \left(\frac{2 \pi t}{32}\right)+18$ , at $t=19$ s using $\Delta t=0.001$ s.

See Solution

Problem 27323

Which statement about average rate of change is FALSE? a) slope of secant line b) compares changes c) uses $\frac{\Delta f}{\Delta x}=\frac{f\left(x_{2}\right)-f\left(x_{1}\right)}{x_{2}-x_{1}}$ d) exact value impossible.

See Solution

Problem 27324

Find the rate of change of blood pressure $P(t)=100-20 \cos \left(\frac{8 \pi}{3} t\right)$ on $[0.2,0.3]$ . Choices: a) 85.6 b) 140 c) 78.3 d) 128.

See Solution

Problem 27325

Find the average rate of change for $f(x)=3(2)^{2 x}$ on $0 \leq x \leq 2$ . Which statement about it is FALSE?

See Solution

Problem 27326

Calculate the average rate of change of $f(x)=\sqrt{x}$ from $x=4$ to $x=9$ . The answer is $\square$ .

See Solution

Problem 27327

Geben Sie den letzten Schritt der Ableitung $f^{\prime}(x)$ an, wenn $f(x) = x^2$ .

See Solution

Problem 27328

Finde den Differentialquotienten $f^{\prime}(x)$ für $f(x)=x^{2}$ und den letzten Schritt der Berechnung: $2x+h$ .

See Solution

Problem 27329

Thermopack: Erklären Sie $T(0)=20$ und $\lim _{t \rightarrow+\infty} T(t)=40$ in einem Satz. Geben Sie $f'(x)$ für $f(x)=x^{2}$ an. Letzten Schritt von $\lim_{h \rightarrow 0}(2x+h)$ nennen.

See Solution

Problem 27330

Gegeben ist die Funktion $f(x)=x^{2}-8$ . Erklären Sie die Schritte zur Berechnung von $f^{\prime}(4)$ .

See Solution

Problem 27331

Finde die Funktion $F(x)$ , deren Ableitung $f(x)=4 x^{4}+4$ ist.

See Solution

Problem 27332

Find the limit as $x$ approaches infinity of $\frac{1}{\sqrt{x}} \int_{1}^{x} \frac{dt}{\sqrt{t}}$ .

See Solution

Problem 27333

Find the absolute max and min of $f(x)=\frac{1}{x}+\ln x$ on $[0.5, 6]$ . Then graph the function.

See Solution

Problem 27334

Find the absolute max and min of $f(x)=\frac{1}{x}+\ln x$ on $[0.5, 6]$ . Graph the function.

See Solution

Problem 27335

Find the max and min of $f(x)=\frac{1}{x}+\ln x$ on $[0.5, 6]$ . Graph the function and determine values at critical points.

See Solution

Problem 27336

Berechne die Ableitung von $f(x) = \sin(x) \cdot x^{2}$ mit Produkt- oder Kettenregel.

See Solution

Problem 27337

At midnight, your house is at $70^{\circ} \mathrm{F}$ and drops to $50^{\circ} \mathrm{F}$ in 2 hours. How long to reach $40^{\circ} \mathrm{F}$ ? Use $T(t)=T_{s}+(T_{0}-T_{s}) e^{-k t}$ .

See Solution

Problem 27338

1. Evaluate $f(3)$ .
2. Find $x$ such that $f(x)=-1$ .
3. State the range in interval notation.
4. Find AROC on $(2,4)$ and $(4,8)$ .
5. Write the piecewise function for the described graph.

See Solution

Problem 27339

Find the average rate of change (AROC) of the function $f(x)=2 x^{3}-x+1$ on the interval $[-2,1]$ .

See Solution

Problem 27340

Ein Fußball wird mit $80 \frac{\mathrm{km}}{\mathrm{h}}$ senkrecht geschossen. Berechne Zeit bis zum höchsten Punkt, maximale Höhe und Fallzeit. Zeichne die Diagramme.

See Solution

Problem 27341

Given $f(x)=3 x^{2}-5 x+7$ , find $D Q$ , explain finding the derivative, and compute $f^{\prime}(x)$ .

See Solution

Problem 27342

Find $f^{\prime}(x)$ for $f(x)=5 e^{4x-9x}$ .

See Solution

Problem 27343

Analyze pressure variation in a static fluid around a sphere of radius $R$ . Integrate to find pressure force and compare with buoyancy.

See Solution

Problem 27344

Find $\frac{d y}{d x}$ for the equation $5 y^{4}+x^{2 \prime}=1$ .

See Solution

Problem 27345

Find the derivative $\frac{d y}{d x}$ for $y=-3 \tan (-x-4)$ . Choose the correct option from the list.

See Solution

Problem 27346

Find the derivative $\frac{d y}{d x}$ for $y=5^{-2 x+7}$ . Choose the correct option from: a) b) c) d).

See Solution

Problem 27347

Find $\frac{d y}{d x}$ for the relation $2 x^{3}=2 y^{2}+5$ .

See Solution

Problem 27348

Find $\frac{d y}{d x}$ for the equation $e^{x}-\sin (y)=x$ .

See Solution

Problem 27349

Find the derivative $\frac{d y}{d x}$ for the function $y=\tan x \cos x$ .

See Solution

Problem 27350

Find the maximum value of the function $f(x) - 2 \sin x \cos x$ . Options: a) 2, b) no max, c) 1, d) 0.

See Solution

Problem 27351

Find the slope of the tangent to $f(x)=5 e^{x}-2 e^{2 x}$ at $x=1$ . Choices: a) 0 b) -1.19 c) - $e^{2}$ d) -15.96

See Solution

Problem 27352

Find the derivative $\frac{d y}{d x}$ for $y=-\frac{e^{-4 x}}{4}$ . Choose the correct option from: a) $\frac{e^{-4 x}}{4}$ , b) $e^{x}$ , c) $\frac{e^{-4 x}}{16}$ , d) $e^{-4 x}$ .

See Solution

Problem 27353

Find the initial velocity of $v(t)=60\left[1-(0.7)^{t}\right]$ and when acceleration is $3 \mathrm{~m/s}^2$ .

See Solution

Problem 27354

Find the derivative $\frac{d y}{d x}$ for $y=\tan x \cos x$ . Options: a) b) c) d)

See Solution

Problem 27355

Find the slope of the tangent to $f(x)=5 x e^{x}$ at $x=2$ .

See Solution

Problem 27356

Find $f^{\prime}(x)$ for $f(x)=5 e^{4 x-9 x}$ . Options: a) $-45 e^{4 x}$ , b) $20 e^{4 x-9 x}$ , c) $-25 e^{4 x-9 x}$ , d) $5 e^{4 x-9 x}$ .

See Solution

Problem 27357

Find the derivative $\frac{d y}{d x}$ for $y=5^{-2 x+7}$ . Options: a) b) c) d)

See Solution

Problem 27358

Find the derivative $\frac{d y}{d x}$ for the equation $3 x^{2}+3 y^{2}=2$ .

See Solution

Problem 27359

How many hours should a student study to maximize effectiveness $E(t) = t\left(2^{-\frac{t}{10}}\right)$ ?

See Solution

Problem 27360

How many hours should a student study to maximize effectiveness $E(t)=t\left(2-\frac{t}{10}\right)$ ?

See Solution

Problem 27361

Calculez la durée de vie moyenne d'un tube au néon avec $f(t)=0,0002 e^{-0,0002 t}$ pour $t \geq 0$ . Utilisez $\mu=\int_{0}^{\infty} t f(t) d t$ .

See Solution

Problem 27362

Find the second derivative of the function $E(t) = t \left(2^{-t / 10}\right)$ .

See Solution

Problem 27363

Find $t$ that maximizes effectiveness $E(t) = t \cdot 2^{-\frac{t}{10}}$ on a scale of 0 to 6, where $t$ is study hours.

See Solution

Problem 27364

Calculez la durée de vie moyenne $\mu$ d'un tube au néon avec $f(t)=0,0005 e^{-0,0005 t}$ pour $t \geq 0$ .

See Solution

Problem 27365

Find the $x$ -coordinate that could be an extreme value for $f(x)=x^{2} 2^{x}$ . Options: a) 1 b) -1 c) 2 d) 0

See Solution

Problem 27366

Find the point where the tangent to $y=-x e^{2 x}$ is horizontal. Options: a) $(0,0)$ , b) none, c) $(-0.5,0.18)$ , d) $(-1.5,0.07)$ .

See Solution

Problem 27367

Find the tangent line to the curve $x^{2}+2xy-y^{2}+x=5$ at the point $P(3,7)$ using implicit differentiation.

See Solution

Problem 27368

Find the function $f$ given $f^{\prime \prime}(x)=8 x^{3}+5$ , $f(1)=1$ , and $f^{\prime}(1)=4$ .

See Solution

Problem 27369

Find the first and second derivatives of the curve $2 y^{3}+x y-6 x+3=0$ at the point $(1,1)$ .

See Solution

Problem 27370

Find $y^{\prime \prime}$ when $x=0$ for the equation $x y + 5 e^{y} = 5 e$ .

See Solution

Problem 27371

Find the max and min values of $f(x)=\frac{x}{x^{2}-x+4}$ on the interval $[0,6]$ .

See Solution

Problem 27372

Which $x$ -coordinate could be an extreme value for the function $f(x)=x^{2} 2^{x}$ ?

See Solution

Problem 27373

Evaluate the limits and value of the piecewise function $g(x)$ at specific points: 1 and 2.

See Solution

Problem 27374

Given the function $h(x)=4 x^{3}+6 x^{2}-24 x+5$ , find where $h$ is increasing, decreasing, concave up, and concave down. Also, find local min/max values and inflection point coordinates.

See Solution

Problem 27375

Find the tangent line equation for $y=\sin ^{2} x$ at the point $\frac{\pi}{2}$ .

See Solution

Problem 27376

Given the function $f(x)=x^{3}-4 x^{2}-16 x+9$ on $[-4,4]$ , is $f$ continuous there? Find $f^{\prime}(x)$ , $f(-4)$ , $f(4)$ , and if Rolle's theorem applies. If so, find $c$ where $f^{\prime}(c)=0$ .

See Solution

Problem 27377

Find the limit: $\lim _{x \rightarrow 2} \frac{x^{2}+1}{x}$

See Solution

Problem 27378

Find the value of $c$ from the mean value theorem for $f(x)=1+\frac{5}{2} \sqrt{x}$ on $[1,25]$ . Choose from: 3, 9, $\frac{9}{2}$ , $\frac{3}{2}$ , None.

See Solution

Problem 27379

Find the critical point of the function $f(x)=\sqrt[3]{4x-5}$ . Choices: $x=\frac{5}{4}$ , $x=\frac{5}{2}$ , $x=0$ , None, $x=-\frac{5}{4}$ .

See Solution

Problem 27380

Check the continuity of the function defined as:
$f(x)=\begin{cases} x^{2}-1 & -1 \leq x<0 \\ 2x & 0 \leq x<1 \\ 1 & x=1 \\ -2x+4 & 1<x \leq 2 \\ 0 & 2<x \leq 3 \end{cases}$

See Solution

Problem 27381

The graph of $\frac{\sin x}{x}$ has which type of asymptote: (a) vertical $x=0$ , (b) oblique $y=x$ , (c) horizontal $y=0$ , or (d) none?

See Solution

Problem 27382

Find $F'(x)$ if $F(x)=\int_{\cos x^{2}}^{0} \frac{-1}{1-x^{2}} dt$ . Choose from options (a) to (d).

See Solution

Problem 27383

Find the value of $c$ in $[0, \frac{\pi}{3}]$ where $f'(c) = \frac{f(\frac{\pi}{3}) - f(0)}{\frac{\pi}{3} - 0}$ for $f(x) = \cos x$ .

See Solution

Problem 27384

Determine when the particle with position $s(t)=t^{3}-5 t^{2}-8 t$ is speeding up for $t \geq 0$ .

See Solution

Problem 27385

Find the first nonzero term of the Maclaurin series for $f(x) = \tan^{-1}(4x)$ .

See Solution

Problem 27386

Find the first four nonzero terms of the Maclaurin series for $f(x)=8 e^{-3 x}$ and its interval of convergence.

See Solution

Problem 27387

Find the first three nonzero terms of the Maclaurin series for $f(x)=\tan^{-1}(4x)$ and its interval of convergence. Choose the correct power series form.

See Solution

Problem 27388

Find the first four nonzero terms of the Maclaurin series for $f(x)=\ln(1+8x)$ and its interval of convergence.

See Solution

Problem 27389

Find the first four nonzero terms of the Maclaurin series for $f(x)=6 \cosh(3x)$ . The first nonzero term is $\square$ .

See Solution

Problem 27390

Write the power series for $6 \cosh 3x$ in summation form and find its interval of convergence.

See Solution

Problem 27391

Find the Taylor series coefficients for $f(x)=\sqrt{x}$ at $a=36$ and use the first four terms to approximate $\sqrt{40}$ .

See Solution

Problem 27392

Find the Taylor series coefficients for $f(x)=\sqrt[3]{x}$ at $a=64$ and use the first four terms to approximate $\sqrt[3]{59}$ .

See Solution

Problem 27393

A 75 kg skier is on a 25 m hill. What is his speed at the bottom? [Answer: 22.1 m/s]

See Solution

Problem 27394

Approximate $\frac{1}{1.19^{2}}$ using the first four nonzero terms of the Taylor series for $f(x)=(1+x)^{-2}$ .

See Solution

Problem 27395

A spring with a constant of $900 \mathrm{~N/m}$ is compressed $0.8 \mathrm{~m}$ . How high does a 2 kg ball launched go? [Answer: $14.7 \mathrm{~m}$ ]

See Solution

Problem 27396

Find the first three nonzero terms of the Taylor series for $f(x)=\sin x$ centered at $a=\frac{5 \pi}{3}$ .

See Solution

Problem 27397

Express $6 \cosh 3x$ as a power series using summation notation.

See Solution

Problem 27398

A 120-pound skier is going 50 m/s at the bottom of the hill. What is the hill's height? Answer: $127.55 \mathrm{~m}$

See Solution

Problem 27399

A sandbag is dropped from a balloon at 300 m, rising at 13 m/s. How fast is it going when it hits the ground? Answer: $-77.8 \mathrm{~m/s}$ .

See Solution

Problem 27400

Approximate $1.17^{-4/5}$ using the first four nonzero terms of the Taylor series for $f(x)=(1+x)^{-4/5}$ .

See Solution

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337

Calculus

Problem 27301

Population de 33000 habitants, taux de naissance 3%3\%3%, mortalité 1%1\%1%, 220 départs/an. Trouvez P(t)P(t)P(t) et P(10)P(10)P(10), puis le temps pour doubler.

Problem 27302

Berechnen Sie die Fläche zwischen dem Graphen von f(x)=−x2−5x−4f(x)=-x^{2}-5x-4f(x)=−x2−5x−4 und der xxx-Achse im Intervall [−5,0][-5, 0][−5,0].

Problem 27303

A parallel-plate capacitor with 20 cm sides has an electric field increasing at dEdt=5.7×1011 V/m⋅s\frac{d E}{d t}=5.7 \times 10^{11} \mathrm{~V/m \cdot s}dtdE​=5.7×1011 V/m⋅s. Find the instantaneous current (i)(i)(i).

Problem 27304

Calculate the integral ∫−11(x2−1) dx\int_{-1}^{1} (x^{2}-1) \, dx∫−11​(x2−1)dx.

Problem 27305

Berechne die Gesamtfläche zwischen dem Graphen von f(x)=0,5x2−3xf(x) = 0,5x^2 - 3xf(x)=0,5x2−3x und der x-Achse.

Problem 27306

Calculate the integral from -1 to 2 of the function x2−1x^{2} - 1x2−1.

Problem 27307

Find the integral of the constant function f(x)=5f(x)=5f(x)=5. What is the result?

Problem 27308

Find the integral ∫(3x2+4x+2) dx\int(3 x^{2}+4 x+2) \, dx∫(3x2+4x+2)dx. What is the result?

Problem 27309

Evaluate the integral ∫(excos⁡(x))dx\int\left(e^{x} \cos (x)\right) d x∫(excos(x))dx.

Problem 27310

What is the correct notation for the indefinite integral of f(x)f(x)f(x) with respect to xxx? Options: ∂∂f(x)dx\partial \partial f(x) d x∂∂f(x)dx, ∬f(x)dx\iint f(x) d x∬f(x)dx, ∂f(x)dx\partial f(x) d x∂f(x)dx, ∫f(x)dx\int f(x) d x∫f(x)dx.

Problem 27311

Identify the definite integral from these options: ∫f(x)dx\int f(x) d x∫f(x)dx, ∬f(x)dx\iint f(x) d x∬f(x)dx, ∂f(x)dx\partial f(x) d x∂f(x)dx, ∫abf(x)dx\int_{a}^{b} f(x) d x∫ab​f(x)dx.

Problem 27312

Problem 27313

Find the total distance traveled by a car with velocity v(t)=3t2−6t+4v(t)=3 t^{2}-6 t+4v(t)=3t2−6t+4 m/s from t=1t=1t=1 to t=5t=5t=5: ∫15v(t)dt\int_{1}^{5} v(t) dt∫15​v(t)dt.

Problem 27314

A savings account has $500.00\$ 500.00$500.00 at 14%14\%14% continuous interest. How much can be spent on a bike after 3 years?

Problem 27315

Find the rate of change of the curve f(x)=−x3f(x)=-x^{3}f(x)=−x3 from x=0x=0x=0 to x=2x=2x=2. Options: -4, -2, 0, 2, 5.

Problem 27316

Betty deposited \$1,000 in a savings account with 13% continuous interest. How much can she spend on a bike in 1 year?

Problem 27317

Find when the velocity of the particle at x(t)=(t+1)(t−3)3x(t)=(t+1)(t-3)^{3}x(t)=(t+1)(t−3)3 is increasing. Choices: (A) t>3t>3t>3 (B) t<1t<1t<1 (C) 1<t<31<t<31<t<3 (D) t<1t<1t<1 or t>3t>3t>3

Problem 27318

Find the integral of sin⁡2x\sin^{2} xsin2x with respect to xxx: ∫sin⁡2x dx\int \sin^{2} x \, dx∫sin2xdx.

Problem 27319

Estimate the instantaneous rate of change of height h(t)=−16cos⁡(2πt32)+18h(t)=-16 \cos \left(\frac{2 \pi t}{32}\right)+18h(t)=−16cos(322πt​)+18 at t=19t=19t=19 s with Δt=0.001\Delta t=0.001Δt=0.001. Round to one decimal.

Problem 27320

Estimate the instantaneous rate of change of h(t)=−16cos⁡(2πt32)+18h(t)=-16 \cos \left(\frac{2 \pi t}{32}\right)+18h(t)=−16cos(322πt​)+18 at t=19t=19t=19 with Δt=0.001\Delta t=0.001Δt=0.001. Which statement is FALSE about average rate of change?

Problem 27321

Find the slope of the tangent to the function f(x)=−3x2+11x−6f(x)=-3 x^{2}+11 x-6f(x)=−3x2+11x−6 at the point P(−4,98)P(-4,98)P(−4,98).

Problem 27322

Find the instantaneous rate of change of height, h(t)=−16cos⁡(2πt32)+18h(t)=-16 \cos \left(\frac{2 \pi t}{32}\right)+18h(t)=−16cos(322πt​)+18, at t=19t=19t=19 s using Δt=0.001\Delta t=0.001Δt=0.001 s.

Problem 27323

Which statement about average rate of change is FALSE? a) slope of secant line b) compares changes c) uses ΔfΔx=f(x2)−f(x1)x2−x1\frac{\Delta f}{\Delta x}=\frac{f\left(x_{2}\right)-f\left(x_{1}\right)}{x_{2}-x_{1}}ΔxΔf​=x2​−x1​f(x2​)−f(x1​)​ d) exact value impossible.

Problem 27324

Find the rate of change of blood pressure P(t)=100−20cos⁡(8π3t)P(t)=100-20 \cos \left(\frac{8 \pi}{3} t\right)P(t)=100−20cos(38π​t) on [0.2,0.3][0.2,0.3][0.2,0.3]. Choices: a) 85.6 b) 140 c) 78.3 d) 128.

Problem 27325

Find the average rate of change for f(x)=3(2)2xf(x)=3(2)^{2 x}f(x)=3(2)2x on 0≤x≤20 \leq x \leq 20≤x≤2. Which statement about it is FALSE?

Problem 27326

Calculate the average rate of change of f(x)=xf(x)=\sqrt{x}f(x)=x​ from x=4x=4x=4 to x=9x=9x=9. The answer is □\square□.

Problem 27327

Geben Sie den letzten Schritt der Ableitung f′(x)f^{\prime}(x)f′(x) an, wenn f(x)=x2f(x) = x^2f(x)=x2.

Problem 27328

Finde den Differentialquotienten f′(x)f^{\prime}(x)f′(x) für f(x)=x2f(x)=x^{2}f(x)=x2 und den letzten Schritt der Berechnung: 2x+h2x+h2x+h.

Problem 27329

Problem 27330

Gegeben ist die Funktion f(x)=x2−8f(x)=x^{2}-8f(x)=x2−8. Erklären Sie die Schritte zur Berechnung von f′(4)f^{\prime}(4)f′(4).

Problem 27331

Finde die Funktion F(x)F(x)F(x), deren Ableitung f(x)=4x4+4f(x)=4 x^{4}+4f(x)=4x4+4 ist.

Problem 27332

Find the limit as xxx approaches infinity of 1x∫1xdtt\frac{1}{\sqrt{x}} \int_{1}^{x} \frac{dt}{\sqrt{t}}x​1​∫1x​t​dt​.

Problem 27333

Find the absolute max and min of f(x)=1x+ln⁡xf(x)=\frac{1}{x}+\ln xf(x)=x1​+lnx on [0.5,6][0.5, 6][0.5,6]. Then graph the function.

Problem 27334

Find the absolute max and min of f(x)=1x+ln⁡xf(x)=\frac{1}{x}+\ln xf(x)=x1​+lnx on [0.5,6][0.5, 6][0.5,6]. Graph the function.

Problem 27335

Find the max and min of f(x)=1x+ln⁡xf(x)=\frac{1}{x}+\ln xf(x)=x1​+lnx on [0.5,6][0.5, 6][0.5,6]. Graph the function and determine values at critical points.

Problem 27336

Berechne die Ableitung von f(x)=sin⁡(x)⋅x2f(x) = \sin(x) \cdot x^{2}f(x)=sin(x)⋅x2 mit Produkt- oder Kettenregel.

Problem 27337

At midnight, your house is at 70∘F70^{\circ} \mathrm{F}70∘F and drops to 50∘F50^{\circ} \mathrm{F}50∘F in 2 hours. How long to reach 40∘F40^{\circ} \mathrm{F}40∘F? Use T(t)=Ts+(T0−Ts)e−ktT(t)=T_{s}+(T_{0}-T_{s}) e^{-k t}T(t)=Ts​+(T0​−Ts​)e−kt.

Problem 27338

1. Evaluate f(3)f(3)f(3).2. Find xxx such that f(x)=−1f(x)=-1f(x)=−1.3. State the range in interval notation.4. Find AROC on (2,4)(2,4)(2,4) and (4,8)(4,8)(4,8).5. Write the piecewise function for the described graph.

Problem 27339

Find the average rate of change (AROC) of the function f(x)=2x3−x+1f(x)=2 x^{3}-x+1f(x)=2x3−x+1 on the interval [−2,1][-2,1][−2,1].

Problem 27340

Ein Fußball wird mit 80kmh80 \frac{\mathrm{km}}{\mathrm{h}}80hkm​ senkrecht geschossen. Berechne Zeit bis zum höchsten Punkt, maximale Höhe und Fallzeit. Zeichne die Diagramme.

Problem 27341

Population de 33000 habitants, taux de naissance $3\%$ , mortalité $1\%$ , 220 départs/an. Trouvez $P(t)$ et $P(10)$ , puis le temps pour doubler.

Berechnen Sie die Fläche zwischen dem Graphen von $f(x)=-x^{2}-5x-4$ und der $x$ -Achse im Intervall $[-5, 0]$ .

A parallel-plate capacitor with 20 cm sides has an electric field increasing at $\frac{d E}{d t}=5.7 \times 10^{11} \mathrm{~V/m \cdot s}$ . Find the instantaneous current $(i)$ .

Calculate the integral $\int_{-1}^{1} (x^{2}-1) \, dx$ .

Berechne die Gesamtfläche zwischen dem Graphen von $f(x) = 0,5x^2 - 3x$ und der x-Achse.

Calculate the integral from -1 to 2 of the function $x^{2} - 1$ .

Find the integral of the constant function $f(x)=5$ . What is the result?

Find the integral $\int(3 x^{2}+4 x+2) \, dx$ . What is the result?

Evaluate the integral $\int\left(e^{x} \cos (x)\right) d x$ .

What is the correct notation for the indefinite integral of $f(x)$ with respect to $x$ ? Options: $\partial \partial f(x) d x$ , $\iint f(x) d x$ , $\partial f(x) d x$ , $\int f(x) d x$ .

Identify the definite integral from these options: $\int f(x) d x$ , $\iint f(x) d x$ , $\partial f(x) d x$ , $\int_{a}^{b} f(x) d x$ .

Find the total distance traveled by a car with velocity $v(t)=3 t^{2}-6 t+4$ m/s from $t=1$ to $t=5$ : $\int_{1}^{5} v(t) dt$ .

A savings account has $\$ 500.00$ at $14\%$ continuous interest. How much can be spent on a bike after 3 years?

Find the rate of change of the curve $f(x)=-x^{3}$ from $x=0$ to $x=2$ . Options: -4, -2, 0, 2, 5.

Find when the velocity of the particle at $x(t)=(t+1)(t-3)^{3}$ is increasing. Choices: (A) $t>3$ (B) $t<1$ (C) $1<t<3$ (D) $t<1$ or $t>3$

Find the integral of $\sin^{2} x$ with respect to $x$ : $\int \sin^{2} x \, dx$ .

Estimate the instantaneous rate of change of height $h(t)=-16 \cos \left(\frac{2 \pi t}{32}\right)+18$ at $t=19$ s with $\Delta t=0.001$ . Round to one decimal.

Estimate the instantaneous rate of change of $h(t)=-16 \cos \left(\frac{2 \pi t}{32}\right)+18$ at $t=19$ with $\Delta t=0.001$ . Which statement is FALSE about average rate of change?

Find the slope of the tangent to the function $f(x)=-3 x^{2}+11 x-6$ at the point $P(-4,98)$ .

Find the instantaneous rate of change of height, $h(t)=-16 \cos \left(\frac{2 \pi t}{32}\right)+18$ , at $t=19$ s using $\Delta t=0.001$ s.

Which statement about average rate of change is FALSE? a) slope of secant line b) compares changes c) uses $\frac{\Delta f}{\Delta x}=\frac{f\left(x_{2}\right)-f\left(x_{1}\right)}{x_{2}-x_{1}}$ d) exact value impossible.

Find the rate of change of blood pressure $P(t)=100-20 \cos \left(\frac{8 \pi}{3} t\right)$ on $[0.2,0.3]$ . Choices: a) 85.6 b) 140 c) 78.3 d) 128.

Find the average rate of change for $f(x)=3(2)^{2 x}$ on $0 \leq x \leq 2$ . Which statement about it is FALSE?

Calculate the average rate of change of $f(x)=\sqrt{x}$ from $x=4$ to $x=9$ . The answer is $\square$ .

Geben Sie den letzten Schritt der Ableitung $f^{\prime}(x)$ an, wenn $f(x) = x^2$ .

Finde den Differentialquotienten $f^{\prime}(x)$ für $f(x)=x^{2}$ und den letzten Schritt der Berechnung: $2x+h$ .

Gegeben ist die Funktion $f(x)=x^{2}-8$ . Erklären Sie die Schritte zur Berechnung von $f^{\prime}(4)$ .

Finde die Funktion $F(x)$ , deren Ableitung $f(x)=4 x^{4}+4$ ist.

Find the limit as $x$ approaches infinity of $\frac{1}{\sqrt{x}} \int_{1}^{x} \frac{dt}{\sqrt{t}}$ .

Find the absolute max and min of $f(x)=\frac{1}{x}+\ln x$ on $[0.5, 6]$ . Then graph the function.

Find the absolute max and min of $f(x)=\frac{1}{x}+\ln x$ on $[0.5, 6]$ . Graph the function.

Find the max and min of $f(x)=\frac{1}{x}+\ln x$ on $[0.5, 6]$ . Graph the function and determine values at critical points.

Berechne die Ableitung von $f(x) = \sin(x) \cdot x^{2}$ mit Produkt- oder Kettenregel.

At midnight, your house is at $70^{\circ} \mathrm{F}$ and drops to $50^{\circ} \mathrm{F}$ in 2 hours. How long to reach $40^{\circ} \mathrm{F}$ ? Use $T(t)=T_{s}+(T_{0}-T_{s}) e^{-k t}$ .

1. Evaluate $f(3)$ .
2. Find $x$ such that $f(x)=-1$ .
3. State the range in interval notation.
4. Find AROC on $(2,4)$ and $(4,8)$ .
5. Write the piecewise function for the described graph.

Find the average rate of change (AROC) of the function $f(x)=2 x^{3}-x+1$ on the interval $[-2,1]$ .

Ein Fußball wird mit $80 \frac{\mathrm{km}}{\mathrm{h}}$ senkrecht geschossen. Berechne Zeit bis zum höchsten Punkt, maximale Höhe und Fallzeit. Zeichne die Diagramme.

Given $f(x)=3 x^{2}-5 x+7$ , find $D Q$ , explain finding the derivative, and compute $f^{\prime}(x)$ .

Find $f^{\prime}(x)$ for $f(x)=5 e^{4x-9x}$ .

Analyze pressure variation in a static fluid around a sphere of radius $R$ . Integrate to find pressure force and compare with buoyancy.

Find $\frac{d y}{d x}$ for the equation $5 y^{4}+x^{2 \prime}=1$ .

Find the derivative $\frac{d y}{d x}$ for $y=-3 \tan (-x-4)$ . Choose the correct option from the list.

Find the derivative $\frac{d y}{d x}$ for $y=5^{-2 x+7}$ . Choose the correct option from: a) b) c) d).

Find $\frac{d y}{d x}$ for the relation $2 x^{3}=2 y^{2}+5$ .

Find $\frac{d y}{d x}$ for the equation $e^{x}-\sin (y)=x$ .

Find the derivative $\frac{d y}{d x}$ for the function $y=\tan x \cos x$ .

Find the maximum value of the function $f(x) - 2 \sin x \cos x$ . Options: a) 2, b) no max, c) 1, d) 0.

Find the slope of the tangent to $f(x)=5 e^{x}-2 e^{2 x}$ at $x=1$ . Choices: a) 0 b) -1.19 c) - $e^{2}$ d) -15.96

Find the derivative $\frac{d y}{d x}$ for $y=-\frac{e^{-4 x}}{4}$ . Choose the correct option from: a) $\frac{e^{-4 x}}{4}$ , b) $e^{x}$ , c) $\frac{e^{-4 x}}{16}$ , d) $e^{-4 x}$ .

Find the initial velocity of $v(t)=60\left[1-(0.7)^{t}\right]$ and when acceleration is $3 \mathrm{~m/s}^2$ .

Find the derivative $\frac{d y}{d x}$ for $y=\tan x \cos x$ . Options: a) b) c) d)

Find the slope of the tangent to $f(x)=5 x e^{x}$ at $x=2$ .

Find $f^{\prime}(x)$ for $f(x)=5 e^{4 x-9 x}$ . Options: a) $-45 e^{4 x}$ , b) $20 e^{4 x-9 x}$ , c) $-25 e^{4 x-9 x}$ , d) $5 e^{4 x-9 x}$ .

Find the derivative $\frac{d y}{d x}$ for $y=5^{-2 x+7}$ . Options: a) b) c) d)

Find the derivative $\frac{d y}{d x}$ for the equation $3 x^{2}+3 y^{2}=2$ .

How many hours should a student study to maximize effectiveness $E(t) = t\left(2^{-\frac{t}{10}}\right)$ ?

How many hours should a student study to maximize effectiveness $E(t)=t\left(2-\frac{t}{10}\right)$ ?

Calculez la durée de vie moyenne d'un tube au néon avec $f(t)=0,0002 e^{-0,0002 t}$ pour $t \geq 0$ . Utilisez $\mu=\int_{0}^{\infty} t f(t) d t$ .

Find the second derivative of the function $E(t) = t \left(2^{-t / 10}\right)$ .

Find $t$ that maximizes effectiveness $E(t) = t \cdot 2^{-\frac{t}{10}}$ on a scale of 0 to 6, where $t$ is study hours.

Calculez la durée de vie moyenne $\mu$ d'un tube au néon avec $f(t)=0,0005 e^{-0,0005 t}$ pour $t \geq 0$ .

Find the $x$ -coordinate that could be an extreme value for $f(x)=x^{2} 2^{x}$ . Options: a) 1 b) -1 c) 2 d) 0

Find the point where the tangent to $y=-x e^{2 x}$ is horizontal. Options: a) $(0,0)$ , b) none, c) $(-0.5,0.18)$ , d) $(-1.5,0.07)$ .

Find the tangent line to the curve $x^{2}+2xy-y^{2}+x=5$ at the point $P(3,7)$ using implicit differentiation.

Find the function $f$ given $f^{\prime \prime}(x)=8 x^{3}+5$ , $f(1)=1$ , and $f^{\prime}(1)=4$ .

Find the first and second derivatives of the curve $2 y^{3}+x y-6 x+3=0$ at the point $(1,1)$ .

Find $y^{\prime \prime}$ when $x=0$ for the equation $x y + 5 e^{y} = 5 e$ .

Find the max and min values of $f(x)=\frac{x}{x^{2}-x+4}$ on the interval $[0,6]$ .

Which $x$ -coordinate could be an extreme value for the function $f(x)=x^{2} 2^{x}$ ?

Evaluate the limits and value of the piecewise function $g(x)$ at specific points: 1 and 2.

Given the function $h(x)=4 x^{3}+6 x^{2}-24 x+5$ , find where $h$ is increasing, decreasing, concave up, and concave down. Also, find local min/max values and inflection point coordinates.